The tables gives
|H1(Γ,G)| for every real form of a simple connected complex
group, except for some intermediate covers in type A.
Also the component groups of the adjoint groups are given.
Here is some detail
about the mathematics, the
realex code used to produce the
tables,
and the output of the script
(H1(Γ,G) for all simple, simply connected groups up to
rank 8).

This page is under construction, October 2013. Please send comments and
corrections
to jda@math.umd.edu.

Simply Connected Classical groups

Group

|H1(Γ,G)|

SL(n,R), GL(n,R), Sp(2n,R)

1

SL(n,H), Spin*(2n)

2

SU(p,q)

⌊p/2⌋
+
⌊q/2⌋+1

Sp(p,q)

p+q+1

Spin(p,q)

⌊(p+q)/4⌋+δ(p,q)

In the last row δ(p,q) depends on p,q mod(4), according to the
following table:

p\q

0

1

2

3

0

3

2

2

2

1

2

1

1

0

2

2

1

1

0

3

2

0

0

0

Simply Connected Exceptional
groups

inner class

group

K

real rank

name

|H1(Γ,G)|

compact

E6

A5A1

4

quasisplitquaternionic

3

E6

D5T

2

Hermitian

3

E6

E6

0

compact

3

split

E6

C4

6

split

2

E6

F4

2

quasicompact

2

compact

E7

A7

7

split

2

E7

D6A1

4

quaternionic

4

E7

E6T

3

Hermitian

2

E7

E7

2

compact

4

compact

E8

D8

8

split

3

E8

A7A1

4

quaternionic

3

E8

E8

0

compact

3

compact

F4

C3A1

4

split

3

F4

B4

1

3

F4

F4

0

compact

3

compact

G2

A1A1

2

split

2

G2

G2

0

compact

2

Special orthgonal groups

group

|H1k(Γ,G)|

SO(p,q)

⌊p/2⌋+⌊q/2⌋+1

SO*(2n)

2

Adjoint classical groups

group

|π0(G(R))|

|H1(Γ,G)|

PSL(n,R)

2

n even

1

n odd

2

n even

1

n odd

PSL(n,H)

1

2

PSU(p,q)

2

p=q

1

otherwise

⌊(p+q)/2⌋+1

PSO(p,q)

1

pq=0

1

p,q odd and p≠q

4

p=q even

2

otherwise

⌊(p+q+2)/4⌋

p,q odd

⌊(p+q)/4⌋+3

p,q even, p+q=0(4)

⌊(p+q)/4⌋+2

p,q even, p+q=2(4)

(p+q+1)/2

p,q opposite parity

PSO*(2n)

2

n even

1

n odd

n/2+3

n even

(n-1)/2+2

n odd

PSp(2n,R)

2

⌊n/2⌋+2

PSp(p,q)

2

p=q

1

otherwise

⌊(p+q)/2⌋+2

In types E8, F4 and G2 the adjoint
group is simply connected.
In simply connected type E6 the center has order 3, and
the adjoint and simply connected groups have the same cohomology (and
the real points are connected). The only essentially new adjoint case is
E7.